How to approximate random motion over one dimensional lattice to Gaussian distribution?

Question

Let us assume a random one dimensional movement where the particle is just allowed to move a unit step in a single time instant. By assuming unbiased random walk ,i.e. , starting from any point, particle can either move in positive or negative direction. Let $P_N(r)$ be the probability that particle reach location r in $N^{th}$ time step derived as: $$P_N(r)=\frac{1}{2^N}\frac{N!}{(\frac{N+r}{2})!(\frac{N-r}{2})!}$$ now it has been said that the given equation for an unbiased case, i.e., when probability of moving in either left or right direction is reduced to Gaussian distribution given by: $$P_N(r) \sim \frac{1}{\sqrt{2\pi N}} \exp({-\frac{r^2}{2N}})$$ using Stirling approximation where $$log N!=NlogN-N+ \frac{1}{2} log(2\pi N)$$ I have tried to do that on my own and able to reduce till: $$log\frac{1}{2^N}+(N+1) log\frac{2N}{\sqrt(N^2-r^2)}-\frac{r}{2} log\frac{N+r}{N-r}-\frac{1}{2} log (2 \pi N)$$ Further, it has been mentioned in question that a factor of 1/2 has been included in the results to amount for the fact r iseven (odd) when N is even (odd).